How Can I Close This Gap Around This Deformed Cylinder?
Some Suggestions .. Not a tutorial
Torus with extrusions, vertex groups, shrink wrap.

Here is a low quality facsimile. I will write some simple things and not enough. Loop tools was shown in another link above.
The yellow cylinder.  The turquoise torus which is modified in edit mode.  So clearly the torus allowed fitting to the cylinder more easily. It has enough vertices for a nice fit. [Extrude] was used to produce the giant extrusion to the right. Then [Loop Cut] was performed to add the grid of vertices so that area can be formed.
The red mark on the right show 5 edges meeting like a box corner meeting a floor, produced by extrusion. A ring was selected and a [vertex group] was created.  [Shrink Wrap Modifier] was added with the Cylinder as a target and is depicted above. If you see terms in [] you do not know, you may want to see a tutorial. So now the vertices will conform to the cylinder when they are moved up (grabbed).  [Proportional Edit] is used. One vertex was selected and raised on the z axis. Nearby vertices followed in a smooth edit.
To the left some other vertices were chosen and put in a vertex group.  A shrink wrap modifier was added, with the box as target. This was added to show straightening an area. Clearly you can straighten an area in many different ways.  Note the red mark on the left where 4 edges meet.  To show a modelling [error], an extrusion was not used.  You can see the diamond shape stretched which you might not want.  No loop cuts were used, just to identify areas.  You may only need on shrink wrap.
I left big gaps in the visual .. you can reduce the gaps.  Please inspect and correct and optimize this work.  You can do better work than this.
Finally one area scaled.  Vertex group created on side and a Shrink Wrap.  You can press [apply] for modifiers.